*Originally posted by Dejection*

**Ah that's interesting. A proof could start as follows:
**

Suppose that there was a number that didn't reach 1. Now look at the sequence generated by this number. There must be a lowest number, say x. {Proof that one can find a lower number in the sequence}. This contradicts the minimality of x, contradiction. Every number must reach 1.

The tricky part of that is showing that eventually all numbers greater than 1 will eventually reach a lower number.

You give me an arbitrary number of times an odd number leads to a higher odd number in 2 iterations consecutively, and I can find you one which will do it that many times in a row, which could lead to a small percentage of cases where the number gets much, much bigger before it gets much smaller.

03 -> 10 -> 05

07 -> 22 -> 11 -> 34 -> 17

15 -> 46 -> 23 -> 70 -> 35 -> 106 -> 53

etc, etc, etc

It would seem these numbers take the form of

**k*(2^m) - 1**, for m-1 consecutive pairs iterations which ends up greater and greater in value.

I am convinced, however, that once that cycle ends, that the numbers tend to drop rather quickly afterwards.

In order to disprove the thesis, you would need to find an example where a certain number loops into itself and would therefore repeat in a manner similar to 1, 4, 2, 1, 4, 2...